QUESTION

Engineering Computations One:

Diﬀerential and Integral Calculus

Assignment Two

Due Thursday 5 April 2012

Please ensure that all your working is clearly set out, all pages are stapled

together, and that your name along with the course name appears on the front

page of the assignment. Please place completed assignments in the assignment

box in WT level 1 by 5pm on the due date.

Questions:

1. Evaluate each of the following limits or explain why it does not exist:

lim

lim

lim

x→2

2x

h→0

e

t→2

t

2

+1

x

2

+6x−4

5+h

2

−4

t

2

h

+4

−e

5

lim

lim

lim

x→2

x

4

−16

x−2

x→2

|x−2|

x→0

x−2

|x| e

sin(π/x)

.

2. The current I at time t seconds in a series circuit containing only a resistor

with resistance 10 ohm, an inductor with inductance 0.5henry,anda

steady 12 volt battery connected at time t = 0 is given by the formula

I =

6

5

1 − e

−20t

(this is shown on page 84 of the course manual). Brieﬂy

explain what happens to the current for t ≥ 0.

3. Sketch the graph of a function that satisﬁes all of the given conditions:

(a) f

(−1) = 0, f

(1) does not exist, f

(x) < 0if|x| < 1, f

(x) > 0if

|x| > 1, f(−1) = 4, f(1) = 0, f

(x) < 0ifx =1.

(b) Domain g =(0, ∞), lim

x→0

+ g(x)=−∞, lim

x→∞

g(x)=0,g

(1) =

0, g

(3) = 0, g

(x) < 0if1<x<3, g

(x) < 0ifx<2orx>4,

g

(x) > 0if2<x<4.

4. At what point(s) on the curve y =2(x − cos x) is the tangent parallel to

the line 3x−y = 5? Illustrate by drawing a rough sketch of the curve, the

line, and the tangent(s).

5. Suppose that f and g are diﬀerentiable functions and that F is the function

given by F(x)=f(x)g(x).

(a) Show that F

= f

g +2f

g

+ fg

1

.

(b) Find similar formulas for F

.

6. Find when the function f(x)=

2x

2

and F

+x−1

x

2

+x−2

(4)

is increasing and decreasing.

7. Use calculus to determine whether the graph of f(x)=

cos x

x

is concave

upward or concave downward at x = π.

8. Suppose the position s of a particle at time t is given by s(t)=t tan t for

0 ≤ t ≤

π

3

.

(a) Find the velocity when t =

π

4

.

(b) Is the particle accelerating or decelerating when t =

.

The fact that

d

dx

sec

2

x =

2sinx

cos

3

x

might be useful.

2

π

4

SOLUTION

(a)

(b)

(c)

(d) Left-hand limit

Right-hand limit

Therefore,

Therefore, the limit does not exist.

(e)

(f)

Left-hand limit

Right-hand limit

Therefore,

2.

For

For

Therefore,

,

4.

Given,

Therefore, we have

The tangent is parallel to the line

Its slope

Therefore,

Therefore,

The point is

5.

(a) Given,

By Leibnitz’ rule,

Therefore,

(b)

6. Given,

For critical points,

In the interval

In the interval

Therefore, the function is increasing on and decreasing on .

7. Given,

Therefore,

At the point

Therefore, the function is concave upwards.

- Given,

(a) The velocity is given by

When,

The velocity is

(b) The acceleration is given by

When,

Therefore, the particle is accelerating.

KC33

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